ABSTRACT
This chapter describes singularities in systems of partial differential equations for which the dependent and independent variables are complex. The singularities will be branch points on an appropriate Riemann surface. A general theory is described for singularities in the solution of hyperbolic equations in a complex space variable. For Laplace's equation, the singularities on the real line at later times are of the same type as the initial singularities in the complex plane. Singularities occur in solutions of algebraic equations and partial differential equations, and are important for many problems in mathematical physics. The nature of the singularities for Burgers' equation can be understood also from the implicit solution. The reason for the square-root behavior for the transverse collision is that the differential equation entails some constraints that force the singularity to be special at a transverse collision. The behavior of systems with more than two speeds is an open question.
